Einblick - Graph Theory Algorithms - # Matching cut problems

Complexity and Algorithms for Matching Cut Problems in Graphs Without Long Induced Paths and Cycles

Q: What other structural graph properties, besides being Pt-free or k-chordal, could lead to tractable or intractable cases for matching cut problems

In addition to being Pt-free or k-chordal, other structural graph properties can also impact the tractability of matching cut problems. For example, graphs with specific connectivity properties, such as being biconnected or having low treewidth, may lead to more efficient algorithms for solving matching cut problems. Additionally, graphs with certain symmetry properties or regular structures could also result in tractable cases for matching cut problems. On the other hand, graphs with high degree vertices, dense connectivity, or complex cycles may lead to more challenging or intractable instances of matching cut problems.

Q: Can the techniques used in this paper be extended to solve other related problems on graph partitioning or graph cuts

The techniques used in the paper can potentially be extended to solve other related problems on graph partitioning or graph cuts. For instance, similar approaches could be applied to problems like finding minimum cuts in graphs, identifying edge connectivity in networks, or optimizing graph partitioning for specific applications. By adapting the algorithmic strategies and reduction techniques employed in the paper, it may be possible to address a broader range of graph partitioning and graph cut problems efficiently.

Q: How do the complexities of matching cut problems compare to the complexity of finding maximum matchings or perfect matchings in these restricted graph classes

The complexities of matching cut problems, such as determining the existence of a matching cut, perfect matching cut, or disconnected perfect matching in graphs, can vary based on the graph classes considered. In comparison to finding maximum matchings or perfect matchings in these restricted graph classes, matching cut problems often involve additional constraints and considerations related to edge cuts and connectivity. As a result, the complexities of matching cut problems may differ from those of finding maximum matchings or perfect matchings, especially in graphs with specific structural properties like being Pt-free or k-chordal.

Kernkonzepte

Matching cut, perfect matching cut, and disconnected perfect matching problems are NP-complete in graphs without induced paths of length 14 or longer, and can be solved in polynomial time in 4-chordal graphs.

Zusammenfassung

The paper studies the computational complexity of three related problems on matching cuts in graphs:

Matching cut (mc): Deciding if a given graph has a matching cut.
Perfect matching cut (pmc): Deciding if a given graph has a perfect matching cut.
Disconnected perfect matching (dpm): Deciding if a given graph has a disconnected perfect matching.

The main results are:

Hardness results:
- mc, pmc, and dpm are NP-complete in {3P6, 2P7, P14}-free 8-chordal graphs.
- Under the Exponential Time Hypothesis (ETH), there is no 2^o(n) time algorithm for these problems on n-vertex {3P6, 2P7, P14}-free 8-chordal graphs.
Positive results:
- dpm and pmc can be solved in polynomial time when restricted to 4-chordal graphs.

The hardness results unify and improve upon previous hardness results for these problems. The polynomial-time algorithms for 4-chordal graphs partially answer an open question on the complexity of pmc in k-chordal graphs.

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Complexity and algorithms for matching cut problems in graphs without long induced paths and cycles

by Hoang-Oanh L... um arxiv.org 04-10-2024

https://arxiv.org/pdf/2307.05402.pdf

Complexity and algorithms for matching cut problems in graphs without long induced paths and cycles

Tiefere Fragen

What other structural graph properties, besides being Pt-free or k-chordal, could lead to tractable or intractable cases for matching cut problems

In addition to being Pt-free or k-chordal, other structural graph properties can also impact the tractability of matching cut problems. For example, graphs with specific connectivity properties, such as being biconnected or having low treewidth, may lead to more efficient algorithms for solving matching cut problems. Additionally, graphs with certain symmetry properties or regular structures could also result in tractable cases for matching cut problems. On the other hand, graphs with high degree vertices, dense connectivity, or complex cycles may lead to more challenging or intractable instances of matching cut problems.

Can the techniques used in this paper be extended to solve other related problems on graph partitioning or graph cuts

The techniques used in the paper can potentially be extended to solve other related problems on graph partitioning or graph cuts. For instance, similar approaches could be applied to problems like finding minimum cuts in graphs, identifying edge connectivity in networks, or optimizing graph partitioning for specific applications. By adapting the algorithmic strategies and reduction techniques employed in the paper, it may be possible to address a broader range of graph partitioning and graph cut problems efficiently.

How do the complexities of matching cut problems compare to the complexity of finding maximum matchings or perfect matchings in these restricted graph classes

The complexities of matching cut problems, such as determining the existence of a matching cut, perfect matching cut, or disconnected perfect matching in graphs, can vary based on the graph classes considered. In comparison to finding maximum matchings or perfect matchings in these restricted graph classes, matching cut problems often involve additional constraints and considerations related to edge cuts and connectivity. As a result, the complexities of matching cut problems may differ from those of finding maximum matchings or perfect matchings, especially in graphs with specific structural properties like being Pt-free or k-chordal.